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    "Imagine we are measuring voltages of a sine wave, and we want to constrain the amplitude and phase.\n",
    "\n",
    "Our measurements have Gaussian noise of known standard deviation, and occur at known times (assume we can measure time so well that there is practically no uncertainty)."
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    "# \"y\" are our measurements\n",
    "y = Array([-1.64, 4.63, 6.09, 5.05, 2.66, -0.90, -6.09, -7.16, -2.54, -1.84])\n",
    "# \"noise_std\" is the standard deviation of the Gaussian noise on our measurements\n",
    "noise_std = 1.0\n",
    "\n",
    "nsamples = length(y)\n",
    "# \"times\" are when the measurements were taken (evenly spaced)\n",
    "times = collect(LinRange(0.0, 2.0*pi, nsamples));"
   ]
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  {
   "cell_type": "code",
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   "source": [
    "using Plots"
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    "Now, we have to write our *model* function.  Our model is that this is a sine wave with parameters of *phase offset* and *amplitude*."
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    "function sine_model(t, phase, amplitude)\n",
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   "source": [
    "tplot = collect(LinRange(0.0, 2.0*pi, 200));\n",
    "# I'm going to guess at parameters: phase = 0, amplitude = 6\n",
    "plot(tplot, sine_model(tplot, 0., 6))\n",
    "plot!(times, y)"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, in order to use MCMC, we must write down our *likelihood*.  Given model parameters, how probable are the measurements we made?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "function sine_log_likelihood(t, y, phase, amplitude)\n",
    "    ymodel = sine_model(t, phase, amplitude)\n",
    "    log_likelihood = -0.5 * sum((y - ymodel).^2 / noise_std.^2)\n",
    "    return log_likelihood\n",
    "end;"
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  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-6.261172416426372"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "sine_log_likelihood(times, y, 0., 6.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-5.560820776311154"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "sine_log_likelihood(times, y, 0.05, 6.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, since this is a low-dimensional problem, we can just grid up the parameter space and compute the likelihood at each position!  In real problems, you can't do this...."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plo,phi = 0., 0.5\n",
    "alo,ahi = 5., 8.\n",
    "pp = range(plo, stop=phi, length=100)\n",
    "aa = range(alo, stop=ahi, length=100)\n",
    "LL = zeros((length(pp), length(aa)))\n",
    "for i in 1:length(pp)\n",
    "    for j in 1:length(aa)\n",
    "        LL[i, j] = sine_log_likelihood(times, y, pp[j], aa[i])\n",
    "    end\n",
    "end\n",
    "heatmap(pp, aa, LL, xlabel=\"Phase\", ylabel=\"Amplitude\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "heatmap(pp, aa, exp.(LL), xlabel=\"Phase\", ylabel=\"Amplitude\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we're going to code up MCMC to see if we can sample this parameter space and get the same answer!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we need to go from a log-likelihood to a log-posterior.  This *requires* specifying a *prior* on the parameters.\n",
    "\n",
    "A common thing to do is to choose a \"flat\" prior: the prior is constant over the entire range.  This is technically wrong (\"improper prior\")."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "function sine_log_posterior(t, y, phase, amplitude)\n",
    "    log_like = sine_log_likelihood(t, y, phase, amplitude)\n",
    "    # Flat prior!  No preference for phase, amplitude values.\n",
    "    log_prior = 0.\n",
    "    return log_like + log_prior\n",
    "end;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the actual MCMC algorithm!  We need to choose a \"proposal distribution\" -- how to select the next candidate parameters to jump to.  We will make these Gaussians with fixed variances."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "chain = []\n",
    "\n",
    "# initial parameter guess\n",
    "param_phase = 0.\n",
    "param_amp = 6.\n",
    "\n",
    "# proposal jump sizes\n",
    "jump_phase = 0.01\n",
    "jump_amp = 0.1\n",
    "\n",
    "# initial log-posterior.\n",
    "logprob = sine_log_posterior(times, y, param_phase, param_amp)\n",
    "\n",
    "# how many steps to take\n",
    "nsteps = 1_000\n",
    "\n",
    "for i in 1:nsteps\n",
    "    \n",
    "    # propose new parameter values\n",
    "    param_phase_new = param_phase + randn() * jump_phase\n",
    "    param_amp_new   = param_amp   + randn() * jump_amp\n",
    "\n",
    "    # compute log-posterior at new parameters\n",
    "    logprob_new = sine_log_posterior(times, y, param_phase_new, param_amp_new)\n",
    "\n",
    "    if (exp(logprob_new - logprob) >= rand(Float64))\n",
    "        logprob = logprob_new\n",
    "        param_phase = param_phase_new\n",
    "        param_amp   = param_amp_new\n",
    "    end\n",
    "    append!(chain, (param_phase, param_amp))\n",
    "end\n",
    "# append! makes \"chain\" a 1-d vector; reshape to a matrix\n",
    "chain = reshape(chain, (2,Int64(length(chain)/2)))';"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1000×2 LinearAlgebra.Adjoint{Any,Array{Any,2}}:\n",
       " -0.00109936   5.98017\n",
       " -0.0115605    6.00415\n",
       " -0.00408136   5.93793\n",
       " -0.00769981   6.00422\n",
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       " -0.00823907   5.90523\n",
       " -0.00699841   5.79549\n",
       " -0.00171511   5.74948\n",
       " -0.0091775    5.55654\n",
       " -0.00535269   5.54681\n",
       " -0.0172107    5.71545\n",
       " -0.0139694    5.72688\n",
       " -0.0139694    5.72688\n",
       "  ⋮                   \n",
       "  0.0228467    6.42657\n",
       "  0.0304117    6.34494\n",
       "  0.0249938    6.38615\n",
       "  0.0232388    6.37255\n",
       "  0.0101356    6.42913\n",
       "  0.0236695    6.46813\n",
       "  0.0143772    6.26921\n",
       "  0.0162767    6.24585\n",
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       "  0.0138851    6.20411\n",
       "  0.0138851    6.20411\n",
       " -0.000254158  6.14351"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "chain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
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       "  1560.84,265.85 1555.55,260.308 1502.95,258.502 1486.82,357.177 1399.57,439.885 1501.83,428.223 1586.5,465.833 1593.08,531.682 1663.95,637.282 \n",
       "  1632.09,647.812 1589.03,511.127 1511.32,454.168 1559.36,498.833 1519.96,535.751 1605.98,558.955 1547.94,595.46 1572.26,648.525 1621.55,620.837 1607.48,578.059 \n",
       "  1577.76,560.285 1612.57,638.679 1734.89,779.647 1763.35,774.669 1712.57,764.431 1699.89,784.687 1688.64,741.936 1644.23,673.644 1642.13,723.568 1702.71,778.634 \n",
       "  1678.85,851.266 1617.03,920.915 1548.45,897.638 1473.83,773.137 1499.1,736.888 1490.04,657.134 1494.83,750.969 1537.9,733.895 1525.82,743.813 1605.24,821.192 \n",
       "  1644.1,872.074 1657.15,842.968 1652.7,852.355 1626.44,889.295 1596.5,945.417 1609.92,888.177 1588.15,922.959 1532.22,1004.27 1478.98,1110.09 1478.65,1101.34 \n",
       "  1437.66,1068.47 1437.62,1120.77 1472.74,1105.75 1369.14,1081.6 1336.31,1099.27 1356.59,1158.48 1274.61,1167.66 1186.38,1213.3 \n",
       "  1129.27,1254.95 1138.94,1130.4 1182.51,1065.34 1158.78,1132.96 1087.78,1171.51 991.549,1167.9 959.793,1172.29 970.134,1121.9 \n",
       "  973.614,1045.14 953.782,1029.5 949.885,992.457 997.726,890.932 965.581,896.943 985.471,846.467 1083.89,911.532 1057.64,892.15 1121.27,965.155 1053.55,1092 \n",
       "  1086.21,948.701 1018.25,975.841 1011.02,979.111 1000.3,896.972 1137.51,879.341 1254.87,878.278 1219.66,869.953 1116.37,757.344 1192.86,771.697 \n",
       "  1163.33,809.186 1154.58,855.84 1080.25,954.208 1103.7,948.979 1084.91,972.244 1075.88,899.841 1097.12,880.781 1138.17,827.932 1144.85,929.075 1076.49,891.631 \n",
       "  1030.35,1015.78 1059.52,979.291 1017.63,970.91 1014.71,973.846 870.227,876.294 909.325,873.156 890.882,810.802 886.952,838.064 926.773,824.208 949.206,863.95 \n",
       "  936.42,873.122 968.693,816.845 944.743,903.517 836.313,1001.19 968.899,961.362 995.202,984.267 1097.63,1017.76 989.11,969.734 900.259,1060.15 \n",
       "  966.36,955.321 940.194,934.984 945.431,891.029 950.506,858.748 1055.63,824.482 1049.87,868.266 1048.34,888.46 1052.52,871.038 1009.17,914.784 1109.98,844.622 \n",
       "  1192.04,764.571 1226.28,884.517 1272.59,761.299 1284.23,823.023 1269.26,871.488 1281.25,893.591 1366.78,829.99 1336.96,824.123 1278.06,883.613 1261.31,824.057 \n",
       "  1331.25,828.524 1326.78,910.863 1283.96,816.039 1288.43,683.711 1223.1,842.447 1240.95,808.215 1150.56,771.352 1191.42,735.216 1195.3,708.853 1178.86,700.939 \n",
       "  1196.93,694.372 1208.13,720.187 1137.59,790.28 1149.29,848.941 1156.47,979.532 1152.22,878.555 1141.9,877.996 1142.13,819.455 1190.03,673.986 \n",
       "  1246.4,621.79 1242.97,618.925 1171.23,724.673 1169.7,786.742 1159.95,830.865 1076.96,716.348 1123.61,745.477 1101.5,723.909 998.394,652.688 \n",
       "  1003.97,577.399 1015.08,547.476 1024.81,579.48 1053.59,597.95 1137.97,488.17 1189.77,522.866 1146.98,560.904 1107.96,540.377 \n",
       "  1161.45,483.217 1162.33,577.826 1236.63,599.974 1207.36,578.374 1203.2,526.182 1213.88,491.731 1178.22,527.016 1169.46,458.597 1201.29,438.718 \n",
       "  1171.57,534.062 1165.37,620.82 1161.13,599.367 1177.65,583.782 1091.15,622.882 1131.25,617.69 1167.32,674.059 1044.29,762.798 1009.5,736.664 \n",
       "  1040.69,728.95 926.518,720.813 789.181,748.724 806.159,666.296 836.545,578.709 824.976,616.681 824.447,534.414 847.009,543.142 834.748,517.773 1004.42,553.391 \n",
       "  1092.45,541.74 1132.79,621.386 1124.58,588.939 1128.81,631.768 1166.04,674.141 1227.86,767.832 1218.51,654.717 \n",
       "  1176.34,614.98 1159.73,697.797 1174.85,684.02 1210.01,682.686 1190.97,567.525 1235.01,695.651 1332.91,675.706 1286.83,689.874 1321.91,773.498 1262.83,894.452 \n",
       "  1213.19,1011.8 1168.12,1077.83 1111.54,1193.7 1090.96,1069.18 1166.22,1019.22 1153.57,984.938 1060.88,1028.01 1136.07,980.76 1159.18,1035.04 \n",
       "  1155.4,1040.14 1089.52,969.784 1115.02,1011.79 1225.35,997.82 1230.13,982.906 1306.99,1023.42 1231.11,1003.47 1286.4,994.747 1182.98,1031.09 1173.61,865.982 \n",
       "  1199.37,834.811 1126.42,815.196 1128.77,914.713 1233.25,914.982 1088.96,894.467 1153.17,949.055 1089.68,791.785 1185.07,749.388 1149.56,801.682 \n",
       "  1232.33,845.761 1285.85,776.566 1313.17,892.229 1398.93,962.358 1464.45,1110.57 1512.36,1146.46 1513.88,1242.17 1525.07,1320.86 1473.63,1372 \n",
       "  1502.44,1284.12 1532.91,1295.19 1526.69,1386.61 1465.3,1337.52 1423.89,1384.99 1441.2,1381.31 1494.95,1366.26 1392.46,1346.1 1318.97,1231.35 1326.27,1244.9 \n",
       "  1263.39,1339.58 1245.99,1323.11 1311.54,1339.02 1293.94,1327.62 1306.86,1297.9 1415.4,1222.03 1453.71,1190.02 1482.01,1161.55 1468.64,1207.93 \n",
       "  1508.94,1130.25 1541.48,1199.64 1529.87,1156.39 1499.84,1036.62 1473.84,975.792 1414.55,964.158 1334.07,969.874 1380.87,975.267 1396.45,1019.15 1309.71,1002.98 \n",
       "  1333.07,1047.78 1350,1043.32 1375.92,901.381 1369.29,1052.33 1333.54,1157.64 1386.08,1123.61 1366.24,1065.95 1362.56,1071.97 1331.34,1179.12 \n",
       "  1340.79,1097.83 1332.59,1004.83 1255.27,938.231 1244.31,940.797 1225.2,1011.84 1117.18,1142.31 1087.87,1116.96 1066.59,1084.82 1050.29,1060.21 1045.41,1016.73 \n",
       "  1104.2,1100.11 1068.29,1097.78 1055.88,1077.14 1118.84,1129.96 1052.55,1111.67 1039,1147.96 1033.87,1099.15 1036.08,1165.48 944.184,1237.04 \n",
       "  920.985,1109.29 936.209,1056.59 913.812,1015.99 918.896,975.941 965.108,1013.72 1004.04,1052.14 933.209,1009.05 936.391,1064.22 945.473,948.341 \n",
       "  910.107,1026.28 945.565,963.234 904.255,953.076 895.378,967.671 944.277,866.564 \n",
       "  929.07,890.39 945.317,727.828 1002.11,813.739 997.18,825.585 1060.59,870.257 1039.66,908.604 902.842,938.354 931.797,851.991 923.548,934.693 \n",
       "  923.691,819.986 950.031,809.668 943.347,777.435 937.487,694.037 1008.61,699.045 1054.13,690.056 915.359,683.081 992.898,782.718 943.353,805.273 \n",
       "  925.105,711.162 964.449,669.084 980.494,637.767 1002.36,624.977 1093.71,675.583 1192.72,728.019 1221.06,777.034 1213.92,805.375 1199.78,798.061 \n",
       "  1134.91,732.399 1161.64,804.168 1176.93,788.944 1174.48,822.9 1132.93,730.005 1112.85,684.523 1071.89,769.188 1033.67,747.675 1006.8,681.466 \n",
       "  1075.55,693.85 1108.81,708.186 1041.47,638.006 1016.85,613.756 1098.24,583.887 1187.61,659.861 1069.36,660.74 1038.48,687.923 1125,630.743 1091.21,713.357 \n",
       "  921.017,674.495 1009.94,713.611 1030.36,745.198 1102.4,730.678 1054.37,637.94 1088.45,612.541 1201.17,682.101 1164.76,774.175 1285.72,810.398 1173.01,863.059 \n",
       "  1123.42,879.073 1105.58,789.542 1064.37,743.197 1030.72,772.584 1112.58,698.752 1164.31,792.008 1141.2,772.642 1079.31,782.488 1117.96,813.956 1162.99,895.63 \n",
       "  1135.3,944.681 1178.13,911.936 1120.5,1006.19 1098.09,930.039 1063.99,857.924 1074.03,805.262 1110.16,768.956 1136.28,705.856 1089.39,734.513 \n",
       "  1038.06,712.103 974.625,679.057 969.24,608.035 1074,598.18 1031.52,660.987 1036.83,649.207 1037.33,663.743 1008,632.45 \n",
       "  1084.21,599.109 1034.84,475.508 1081.08,385.631 1136.8,302.145 1142.24,324.445 1103.88,290.454 1094.28,302.796 1085.11,144.741 1013.05,158.811 971.45,86.2547 \n",
       "  1086.61,144.615 1089.72,200.867 1066.42,258.164 1091.51,263.79 1175.39,309.769 1238.19,345.505 1246.58,357.473 1233.73,446.25 \n",
       "  1233.83,472.7 1259.36,467.457 1248.03,454.945 1190.33,449.368 1224.51,396.113 1311.24,379.382 1306.78,284.958 1368.35,228.896 1393.59,254.535 1418.29,272.434 \n",
       "  1499.88,449.615 1475.91,444.374 1435.99,449.514 1414.05,432.136 1386.18,496.443 1444.55,527.807 1450.44,612.218 1439.53,569.907 1463.71,606.198 \n",
       "  1501.5,678.198 1489.79,657.261 1579.47,732.874 1543.83,768.643 1630.47,842.154 1599.1,822.436 1661.92,792.238 1780.2,691.92 1785.15,724.46 \n",
       "  1805.63,760.497 1719.87,743.759 1723.21,842.102 1797.42,805.203 1855.74,810.468 1899.6,757.995 1849.86,822.77 1885.83,766.753 \n",
       "  1854.18,706.776 1939.19,773.258 1902.89,756.986 1964.68,756.743 1915.85,743.372 1904.69,756.36 1944.88,732.841 1963.38,921.12 2008.22,909.611 \n",
       "  2139.5,791.236 2250.68,683.854 2282.79,647.878 2254.45,690.203 2292.27,663.54 2287.63,704.16 2217.27,715.248 2143.97,697.938 2103.93,677.73 \n",
       "  2055.04,635.464 2073.75,554.63 2040.17,604.224 2006.25,595.849 1881.1,569.912 1889.24,563.343 1839.68,463.716 1759.22,556.908 1688.24,517.581 \n",
       "  1769.93,613.24 1808.95,503.911 1745.3,564.503 1733.53,576.512 1840.75,622.621 1872.05,609.972 1815.61,651.773 1822.64,637.242 \n",
       "  1875.35,627.319 1882.45,615.899 1814.04,523.664 1834.08,538.767 1842.51,495.871 1890.78,549.046 1875.15,556.368 1729.95,661.811 1688.8,591.115 \n",
       "  1721.9,620.048 1700.8,670.32 1694.89,742.154 1714.55,710.913 1693.42,669.212 1719.08,694.479 1670.85,610.972 1764.59,654.697 1828.07,638.614 \n",
       "  1741.75,567.949 1803.17,511.151 1823.5,502.851 1841.79,464.064 1851.91,553.135 1819.9,615.334 1866.77,582.055 \n",
       "  1841.75,588.323 1837.4,593.652 1846.67,583.916 1860.41,624.115 1766.58,698.393 1734.55,763.662 1637.55,781.279 1590.22,740.727 \n",
       "  1672.6,796.003 1690.26,779.789 1696.48,783.04 1683.03,682.974 1832.61,729.678 1792.68,693.404 1824.06,693.949 1844.42,652.599 \n",
       "  1834.2,620.414 1883.43,816.95 2009.8,781.115 1977.8,792.38 1928.62,920.401 1948.75,904.899 1895.11,998.437 1890.97,1013.11 \n",
       "  1789.63,931.778 1790.69,914.008 1857.38,939.991 1793.78,849.325 1826.89,996.659 1819.86,953.245 1787.49,1065.43 1796.39,1010.54 1724.03,1022.26 1741.67,915.38 \n",
       "  1735.08,1021.67 1777.96,904.361 1835.64,987.204 1897.98,973.112 1924.25,1004.58 1858.02,1051.99 1897.07,1061.81 1873.16,1070.11 1916.52,1141.91 1951.1,1215.58 \n",
       "  1841.74,1219.63 1774.63,1238.01 1800.35,1287.45 1845.19,1271.93 1804.27,1220.43 1825.04,1179.93 1839.26,1025.2 1808.56,996.45 1812.22,1009.5 1724.01,918.983 \n",
       "  1633.64,1006.07 1699.17,1048.39 1661.08,930.784 1623.92,893.141 1635.77,808.94 1613,721.266 1531.85,691.749 1499.13,786.57 1547.98,792.264 \n",
       "  1599.52,790.699 1528.35,790.303 1515.16,892.906 1488.78,904.647 1549.59,877.445 1487.38,953.542 1520.31,901.185 1545.83,780.537 1607.61,870.105 \n",
       "  1626.48,881.995 1673.35,779.999 1654.75,707.974 1592.43,773.958 1546.18,800.476 1568.99,857.755 1425.69,808.433 1387.27,764.261 1347.69,758.163 \n",
       "  1330.69,744.802 1352.89,773.822 1437.88,530.721 1404.01,472.166 1344.03,513.76 1338.5,453.303 1302.59,457.707 1338.38,462.459 1326.03,527.927 \n",
       "  1274.01,628.902 1264.82,600.019 1302.24,598.907 1392.81,488.274 1411.64,533.498 1423.49,514.789 1334.26,496.592 1321.84,462.696 1296.59,499.508 \n",
       "  1410.67,481.538 1499.42,518.185 1525.2,494.287 1444.11,475.203 1491.98,519.929 1558.52,578.826 1552.53,557.932 1540.37,635.545 1519.6,708.366 \n",
       "  1536.51,687.381 1568.93,650.423 1571.35,647.244 1598.76,634.254 1590.23,630.517 1602.76,671.144 1592.81,610.933 1656.87,650.703 1622.77,597.848 \n",
       "  1698.95,597.856 1711.08,578.579 1672.86,552.844 1683.09,639.124 1704.44,703.937 1785.42,774.966 1751.46,764.173 1812.32,622.649 1853.18,675.383 1821.68,741.369 \n",
       "  1746.32,742.49 1760.37,782.351 1765.48,796.766 1779.04,758.504 1734.39,715.05 1690.19,870.862 1658.56,847.984 1673.42,864.937 \n",
       "  1650.94,982.289 1543.03,963.8 1420.79,923.132 1457.41,952.519 1449.05,1026.58 1401.01,980.275 1386.82,950.708 1315.6,1033.88 1336.53,1010.72 \n",
       "  1405.64,973.298 1427.1,1038.64 1378.53,1056.79 1281.13,928.329 1289.12,873.404 1130.4,856.8 1217.4,929.008 1159.72,866.36 1175.25,917.832 1285.98,892.995 \n",
       "  1250.09,979.341 1185.49,993.344 1124.24,968.011 1043.97,974.421 1077.91,980.398 1004.52,946.897 952.379,916.466 936.024,969.996 917.484,909.792 \n",
       "  929.8,995.596 854.949,972.552 847.021,923.833 834.797,898.992 810.471,748.061 760.493,715.624 769.281,704.742 788.565,697.295 803.911,710.358 805.926,674.324 \n",
       "  822.018,611.752 757.813,648.279 717.191,636.846 656.469,588.848 552.9,544.402 686.167,555.205 649.164,557.022 745.151,526.964 \n",
       "  679.459,535.218 743.5,641.869 769.215,553.933 855.338,585.563 902.314,649.43 940.495,672.528 801.269,652.454 889.34,739.113 972.205,762.499 984.813,684.39 \n",
       "  1012.96,668.803 995.041,706.856 1024.6,635.827 952.516,512.042 978.698,609.796 967.715,567.126 971.287,594.597 923.32,629.711 884.848,605.127 \n",
       "  828.293,593.833 748.472,612.397 679.278,519.766 642.181,523.882 670.973,595.316 693.736,625.594 704.494,636.732 755.23,651.373 726.992,584.086 760.094,558.194 \n",
       "  728.331,648.327 738.799,683.948 766.306,663.783 808.998,736.218 752.25,719.258 735.582,706.713 713.91,775.172 650.647,783.968 \n",
       "  700.001,714.245 725.202,590.325 774.928,596.538 849.188,604.459 811.77,586.952 729.402,621.675 745.3,677.897 715.745,774.559 691.13,832.166 766.924,901.551 \n",
       "  785.796,916.055 820.376,885.665 793.12,853.225 771.762,769.154 728.772,733.486 638.876,666.219 666.676,701.193 710.451,753.371 \n",
       "  679.1,727.027 668.945,735.726 593.122,699.553 671.437,674.626 617.666,801.783 628.658,816.713 539.849,885.934 614.819,843.398 533.001,882.134 \n",
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     "execution_count": 43,
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   ],
   "source": [
    "plot(chain[:,1], chain[:,2], xlabel=\"Phase\", ylabel=\"Amplitude\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
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